A new approach to BVPs with state-dependent impulses
© Rachůnková and Tomeček; licensee Springer. 2013
Received: 24 October 2012
Accepted: 15 January 2013
Published: 11 February 2013
The paper deals with the second-order Dirichlet boundary value problem with one state-dependent impulse
Proofs of the main results contain a new approach to boundary value problems with state-dependent impulses which is based on a transformation to a fixed point problem of an appropriate operator in the space . Sufficient conditions for the existence of solutions to the problem are given here. The presented approach can be extended to more impulses and to other boundary conditions.
Most papers in the literature on impulsive boundary value problems concern the case with fixed moments of impulsive effects. Papers dealing with state-dependent impulses, called also impulses at variable times, focus their attention on initial value problems or periodic problems. Such papers investigate the existence, stability or asymptotic properties of solutions of initial value problems [4–8] or solvability of autonomous periodic problems [9, 10] and nonautonomous ones [11–15]. We can also find papers investigating other boundary value problems with state-dependent impulses through some initial value problems for multi-valued maps [16, 17].
Under assumptions (4)-(8), we prove the solvability of problem (1)-(3). In particular, we transform problem (1)-(3) to a fixed point problem for a proper operator in the space . This approach can be also used for other types of boundary conditions and it can be easily extended to more impulses.
It is well-known that the mentioned normed spaces are Banach spaces. Recall that for , a function satisfies the Carathéodory conditions on (we write ) if
is measurable for all ,
is continuous for a.e. ,
for each compact set , there exists a function such that for a.e. and each .
We say that is a solution of problem (1)-(3), if z is continuous on , there exists unique such that , and have absolutely continuous first derivatives, z satisfies equation (1) for a.e. and fulfills conditions (2), (3).
Therefore, σ is strictly decreasing on and hence it has exactly one root in . □
where fulfills (9). The next lemma provides an important result about the continuity of which is fundamental for our approach.
Lemma 2 The functional is continuous on .
for each . By Lemma 1 and the continuity of , it follows that for . □
Lemma 3 The operator ℱ is compact on .
as uniformly w.r.t. . Therefore, converges to x in . Similar arguments can be applied to the sequence .
holds for all . Consequently, is relatively compact in by the Arzelà-Ascoli theorem. □
is a solution of problem (1)-(3).
Therefore, σ is strictly decreasing on , which yields for . Consequently, is a unique point in satisfying (20).
3 Main result
Here, using the Leray-Schauder degree theory, we prove our main result about the solvability of problem (1)-(3). To this end, we will need the following lemma on a priori estimates.
. Since , it follows that and therefore , , and . There are two possibilities as follows.
Then , which yields .
which is a contradiction.
For , the solution of (21) is , and it clearly belongs to Ω. □
Theorem 6 Assume (4)-(8). Then the operator ℱ has a fixed point in Ω.
has a solution in Ω. This solution is a fixed point of the operator ℱ. □
Proof From Theorem 6 it follows that there exists a fixed point of the operator ℱ. Lemma 4 yields that the function z defined in (17) (with ) is a solution of problem (1)-(3). Estimates (25) follow from (17) and from the definitions of Ω and (cf. (12) and (8)). □
In this section we demonstrate that Theorem 7 can be applied to sublinear, linear and superlinear problems.
Example 9 (Sublinear problem)
we can check that conditions (8) are satisfied in both cases. Therefore, by Theorem 7, the corresponding problem (1)-(3) has at least one solution.
Note that (27) shows that γ need not be monotonous.
Example 10 (Linear problem)
If (28) holds, then for any sufficiently large K, condition (7) is satisfied. By (8), we have , and problem (1)-(3) has a solution for any γ satisfying (8). Consequently, if γ is given by (26) or (27), problem (1)-(3) is solvable.
satisfies (30) as well. Put, for example, , . Then we get that for inequality (30) holds. Consequently, (8) gives and the corresponding problem (1)-(3) is solvable for any γ satisfying (8). In particular, γ given by (26) or (27) can be considered in this case as well.
Dedicated to Jean Mawhin on the occasion of his 70th birthday.
The authors would like to thank the anonymous referees for their valuable comments and suggestions. This work was supported by the grant Matematické modely a struktury, PrF_ 2012_ 017.
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